Spatially heterogeneous stochastic petri-net modeling

ABSTRACT

Modeling of biochemical reactions of a system is accomplished using spatially heterogeneous stochastic Petri-net modeling. The biochemical reactions of the system to be modeled are defined. A spatial decomposition of the system is defined by defining regions of a space in which the system is to be modeled and by assigning each biochemical reaction to a region, such that the system is spatially heterogeneous. Relationships for inter-region movement of reactants of the biochemical reactions are defined as flux, advection, convection, and/or diffusion-based molecular movements. The system of the biochemical reactions is then modeled by modeling the biochemical reactions of each region as a spatially homogenous stochastic Petri-net and by modeling the inter-region movement of the reactants based on the relationships as defined.

FIELD OF THE INVENTION

The present invention relates generally to stochastic Petri-netmodeling, and more specific to modeling spatially heterogeneous systemsusing stochastic Petri-net modeling.

BACKGROUND OF THE INVENTION

To understand the molecular logic of a cell, methods of modeling andsimulation are important. The primary motivation for usingcomputer-aided study mechanisms is that observing biochemical systemsand conducting experiments on them is difficult. Yet, such observationand experimentation can be necessary to improve health care. Forexample, the study of gene regulatory and metabolic networks plays animportant role in the detection of genetic or metabolic defects, as wellas in therapeutic drug research. Genetic and metabolic defects oftenlead to diseases like high blood pressure, Alzheimer's, cancer, anddiabetes. Biochemical models can be used to integrate detailedbiochemical data, and to help understand the behavior of complex systemsof molecular interactions. Building such biochemical models has remainedan art and an arduous task.

Many processes in molecular biology, as well as other biochemicalsystems, demonstrate a stochastic nature. Such biological phenomenanecessitate the use of stochastic models. The primary difference betweendeterministic and stochastic models is that in a deterministic model aninitial condition results in one, and only one, final outcome, whereasin a stochastic model distinct final outcomes can arise from identicalinitial conditions. Therefore, the theoretical understanding ofmolecular and developmental biology using computational systems has toaccount for variations caused by the stochastic interactions ofmolecules and other reactants, as well as the large variety of thesemolecules and the complex feedback loops characterizing biochemicalsystems. Traditional mathematical tools are not well suited to modelingsuch dynamic behavior.

One type of mathematical formalism that has been applied to biochemicalsystems is the Petri-net. The Petri-net formalism is a graphicallyoriented language of design, specification, simulation, and verificationof systems. It offers methods to represent the structure of adiscrete-event system, to simulate the system's behavior, and to drawcertain types of general conclusions regarding the properties of thesystem. Simple Petri-net models, however, do not provide features thatcan capture quantitative aspects of stochastic biochemical systems.

However, the reference P J E Goss and Peccoud, “Quantitative modeling ofstochastic systems in molecular biology using stochastic Petri-nets,” inProceedings of the National Academy of Sciences, USA, vol. 95, pp.6750-6755 (June 1998) [hereinafter Goss and Peccoud], describes anextended Petri-net formalism having features that allow for the modelingof stochastic biochemical systems. Such augmented Petri-net frameworksmay be referred to as spatially homogenous stochastic Petri-netframeworks, and have been successfully applied to the modeling andsimulation of many complex biological and other biochemical systems.

A significant drawback of the spatially homogenous stochastic Petri-netframework described in [Goss and Peccoud] is its inability to representspace heterogeneously, and hence its inability to model effectsresulting from non-uniform spatial distributions of interactingmolecules and other reactants. In other words, current Petri-netframeworks are unable to model systems existing within spatiallyheterogeneous spaces, in which biochemical reactants are non-uniformlyspatially distributed. For this and other reasons, therefore, there is aneed for the present invention.

SUMMARY OF THE INVENTION

The present invention relates to spatially heterogeneous systemsmodeling using stochastic Petri-net. A method of the present inventiondefines one or more biochemical reactions of a system to be modeled.Each biochemical reaction has one or more reactants, such as molecules.The method defines a spatial decomposition of the system, by definingregions of a space in which the system is to be modeled, and byassigning the biochemical reactions to all the regions, and by assigningthe molecular distribution of the reactants of the reactions among theregions, such that the regions are initially populated within theregions in accordance with this distribution. The result is that thesystem is spatially heterogeneous. The method further definesrelationships for inter-region movement of the reactants of thebiochemical reaction. These relationships may be flux, advection,convection, and/or diffusion-based molecular movement. The method modelsthe system of the biochemical reactions by modeling the biochemicalreactions of each region as a spatially homogenous stochastic Petri-net,and by modeling the inter-region movement of the reactants based on therelationships that have been defined, such as based on flux, advection,convection, and/or diffusion-based molecular movement.

A system of the present invention includes one or more processors, oneor more computer-readable media, and a computer program. Thecomputer-readable media may each be a volatile or non-volatile medium,as well as a magnetic, optical, and/or semiconductor medium. The mediastore data representing one or more biochemical reactions and a numberof regions of a space. Each biochemical reaction is assigned to all theregions, but the molecular distribution of the various species presentin the region is varied such that the biochemical reactions arespatially heterogeneous. The computer program is executed by theprocessors to model the biochemical reactions of each region as aspatially homogenous stochastic Petri-net. The computer-readable mediamay further store data representing relationships corresponding topotential inter-region movement of reactants of the biochemicalreactions, such that the computer program may further model inter-regionmovement of the reactants based on these relationships, which may beflux, advection, convection, and/or diffusion-based molecular movement.

An article of manufacture of the present invention includes acomputer-readable medium, and means in the medium. The medium may be avolatile or a non-volatile medium, as well as a magnetic, optical,and/or semiconductor medium, or a modulated carrier signal. The means isfor modeling a system of one or more biochemical reactions as aspatially heterogeneous stochastic Petri-net. The means may model eachbiochemical reaction as a spatially homogeneous stochastic Petri-netwithin a region of a space into which the system has been spatiallyheterogeneously decomposed. The means may further model inter-regionmovement of reactants of the biochemical reactions as flux, advection,convection, and/or diffusion-based molecular movement.

Embodiments of the invention provide for advantages over the prior art.Unlike the spatially homogenous stochastic Petri-net modeling describedin [Goss and Peccoud], the present invention provides for modelingspatially heterogeneous systems using stochastic Petri-nets. The netresult is that more accurate modeling of biochemical systems can beachieved, since the spatially heterogeneous nature of such systems iscaptured within the modeling.

In particular, the present invention provides a methodology by which asspatially homogenous stochastic Petri-nets, can be used to modelspatially heterogeneous systems. A system of biochemical reactions iscompartmentalized by representing and treating each compartment of thesystem, such as each region or orthant, as a homogeneous well-mixedsystem, and using the spatially homogenous stochastic Petri-netformalism to represent the biochemical reactions within the region. Themovement of reactants, such as molecules, across compartments isrepresented as flux, advection, convection, and/or diffusion-basedmolecular movement.

Still other advantages, aspects, and embodiments of the invention willbecome apparent by reading the detailed description that follows, and byreferring to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings referenced herein form a part of the specification.Features shown in the drawing are meant as illustrative of only someembodiments of the invention, and not of all embodiments of theinvention, unless otherwise explicitly indicated, and implications tothe contrary are otherwise not to be made.

FIG. 1 is a diagram of an example and representative biochemical systemmodeled as a spatially homogenous stochastic Petri-net, in conjunctionwith which embodiments of the invention may be practiced.

FIG. 2 is a diagram of the biochemical system of FIG. 1 after thespatially homogenous stochastic Petri-net model has been run, inconjunction with which embodiments of the invention may be practiced.

FIG. 3 is a diagram of the example modeling of a representativebiochemical system, taking into account the spatially heterogeneousnature of the system, according to an embodiment of the invention.

FIG. 4 is a diagram of the example modeling of the representativebiochemical system of FIG. 3, in which the system is instead presumed tobe spatially homogeneous, to demonstrate the advantages of theembodiment of FIG. 3 over the prior art.

FIG. 5 is a flowchart of a method for heterogeneously spatial stochasticPetri-net modeling of a biochemical system, according to an embodimentof the invention.

FIG. 6 is a flowchart of a method to define a biochemical reaction as astochastic Petri-net model, according to an embodiment of the invention.

FIG. 7 is a diagram depicting spatial decomposition of athree-dimensional cubic space into a number of three-dimensional spatialregions, or orthants, according to an embodiment of the invention.

FIG. 8 is a diagram of a rudimentary computerized system forheterogeneously spatial stochastic Petri-net modeling of a biochemicalsystem, according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention. Other embodiments may be utilized,and logical, mechanical, and other changes may be made without departingfrom the spirit or scope of the present invention. The followingdetailed description is, therefore, not to be taken in a limiting sense,and the scope of the present invention is defined only by the appendedclaims.

Technical Background: Spatially Homogenous Stochastic Petri-Nets

FIG. 1 shows a spatially homogenous stochastic Petri-net representationof an example and representative biochemical system 100, in conjunctionwith which embodiments of the invention may be practiced. The variouselements of the biochemical system 100 are represented as differentelements of the stochastic Petri-net. There are places 102A, 102B, and102C, collectively referred to as the places 102. Each place of thestochastic Petri-net represents a molecular species. Each place can havezero or more tokens located thereat, where each token represents anindividual molecule or other reactant of the molecular species that theplace represents. For instance, in FIG. 1 there are tokens 104A, 104B,and 104C, collectively referred to the tokens 104, located at the place102A, and there are tokens 106A and 106B, collectively referred to asthe tokens 106, located at the place 102B. There are no tokens locatedat the place 102C.

There is also in FIG. 1 a transition 110. A transition of the stochasticPetri-net represents a reaction. A reaction is the creation, generation,or transformation of one or more molecules or other reactants from oneor more existing molecules or other reactants. For example, thetransition 110 represents a biochemical reaction in which the reactantsdenoted by R1 are combined with the reactants denoted by R2 to result inthe product denoted by P.

The places 102 and the transition 110 are linked by input/outputfunctions 112B, 112C, and 112D, collectively referred to as theinput/output functions 112. An input/output function implements thestoichiometric constraints governing the reactions being modeled. Eachof the input/output functions 112 is represented as a directed arc. Thefunctions 112 have corresponding weights 114B, 114C, and 114D,collectively referred to as the weights 114. The weights 114 indicatethe stoichiometric requirements for and outputs of the occurrence oftheir corresponding transitions.

For example, the input/output function 112B has a weight 114B of two,and the input/output function 112C has a weight 114C of one, indicatingthat two of the tokens 104 of the place 102A and one of the tokens 106of the place 102B are consumed in the reaction represented by thetransition 110. Furthermore, the input/output function 112D has a weight114D of one, indicating that the reaction represented by the transition110 results in one token being generated within the place 102C.

The transition 110 further has a label 118. A label indicates the numberof time steps that are to occur between successive firings of thetransition to which the label corresponds. For example, the transition110 has a label 118 of zero, indicating that the reaction represented bythe transition 110 is instantaneous, and can occur so long as there aresufficient tokens as required by the weights 114B and 114C of theinput/output functions 112B and 112C leading to the transition 110.

The transition 110 is said to be enabled when all of its input places102A and 102B have a number of tokens greater than or equal to theweights 114B and 114C of the corresponding input/output functions 112Band 112C. Enabled transitions are able to fire, and each firing eventrepresents the occurrence of the biochemical reaction represented by acorresponding transition. The decision as to whether a transition firesor not is probabilistic, or stochastic or random, and is furthergoverned by the label 118 of the transition 110 as has been described.

As has been noted, each of the places 102 is able to hold one or more ofthe tokens 104 and 106. The number of tokens in a place is referred toas the marking of the place. A vector representing the tokens in eachplace at a given moment t is referred as the global marking Mt of thebiochemical system 100. For example, the biochemical system 100 has aglobal marking M={3, 2, 0} at the time depicted in FIG. 1. This meansthat there are three tokens 104 within the place 102A, two tokens 106within the place 102B, and no tokens within the place 102C.

Thus, the stochastic Petri-net representation of the system 100 of FIG.1 has three places, also denoted as the places R1, R2, and P. The placeR1 represents a first reactant, the place R2 represents a secondreactant, and the place P represents the product of the first and thesecond reactants resulting from the biochemical reaction represented bythe transition also denoted by Re1. There is one transition: thetransition denoted by Re1 that represents a reaction involving the firstand the second reactants, and producing the product thereof

The input functions for the reaction represented by the transitiondenoted by Re1 determine the stoichiometric requirements for thereaction. This reaction is:2R ₁ +R ₂ →P.Because the label 118 of the transition 110 is zero, the reactionrepresented by the transition denoted by Re1 is instantaneous, such thatif possible the transaction Re1 fires each time step and consumes onetoken each from the places R1 and R2, producing a token in P1.

FIG. 2 shows an example of the spatially homogenous stochastic Petri-netrepresentation of the example and representative biochemical system 100of FIG. 1 after one or more time steps. The system 100 of FIG. 1 isindicated as the system 100′ in FIG. 2 to differentiate the differentpoints in time that the system 100 is depicted in FIG. 1 as compared toFIG. 2. In FIG. 2, the transition 110 representing a reaction anddenoted by Re1 has fired.

As a result, there is one token within the place 102A also denoted byR1. This is because two of the tokens were consumed by the reaction ofthe transition 110, as required by the weight 114B of the input/outputfunction 112B. Furthermore, there is just one token 106A within theplace 102B also denoted by R2. This is because the other token 106B wasconsumed by the reaction of the transition 110, as required by theweight 114C of the input/output function 112C. Finally, there is onetoken 202 within the place 102C denoted by P. This is because thereaction of the transition 110 resulted in the generation of the token202 at the place 102C, as indicated by the weight 114D of theinput/output function 112D.

It is noted that the biochemical system 100 modeled in FIGS. 1 and 2using spatially homogenous Petri-nets is stochastic. This means thatwhile the transition 110 has fired in the example of FIG. 2, in adifferent running of the system 100, with the same initial conditions asin FIG. 1, the transition 110 may not fire, even though the transition110 is enabled and thus capable of firing. That is, the stochastic, orrandom or probabilistic, nature of the system 100 means that the finaloutcome of modeling the system 100 may be different even with the sameinitial conditions of the system 100.

Furthermore, the biochemical system 100 is modeled in FIGS. 1 and 2using stochastic Petri-nets that are spatially homogenous. This meansthat all of the molecules or other reactants corresponding to a givenspecies can be represented by one of the single places 102, which occursonly where the system 100 is well mixed, and spatially uniform. However,in many types of systems, this spatial homogeneity assumption isinvalid, as molecules and other reactants are distributed unequally ornon-uniformly within a given space, such that the reactions in whichthey participate can vary greatly.

FIGS. 3 and 4 show how presuming that a spatially heterogeneousbiochemical system 300 is spatially homogenous can affect the modelingof the system 300. FIG. 3 shows the results of modeling a spatiallyheterogeneous biochemical system 300 in accordance with an embodiment ofthe invention, whereas FIG. 4 shows the results of modeling the system300 where the system 300 is presumed to be spatially homogenous, suchthat the spatially heterogeneous nature of the system 300 is ignored. InFIG. 3, in other words, modeling of the biochemical system 300 isaccomplished while preserving the spatially heterogeneous, or spatiallynon-uniform, nature of the system 300. By comparison, in FIG. 4,modeling of the biochemical system 300 is accomplished while ignoringthe spatial heterogeneous, or spatially non-uniform, nature of thesystem 300.

First referring to FIG. 3, there are two sets of reactants 302 and 304within the biochemical system 300. The reactants 302 are denoted assolid circles and the reactants 304 are denoted as hollow circles forillustrative clarity. The reactants 302 are heterogeneously, ornon-uniformly, distributed over a number of spatial regions 308A, 308B,. . . , 308N, collectively referred to as the spatial regions 308, asshown in FIG. 3, as are the reactants 304. The reaction of the reactants302 and 304 results in the set of products 306, which are denoted assmall X's in FIG. 3 for illustrative clarity.

The products 306 in FIG. 3 are concentrated in the middle spatialregion, because this is substantially the only region in which there issubstantial overlap between the reactants 302 and the reactants 304. Asa result, even after the products 306 have been generated, there aresignificant numbers of both the reactants 302 and the reactants 304. Forinstance, because there are only reactants 302 in the upper-right andthe lower-left regions, and none of the reactants 304 in these regions,the reactants 302 in these regions remain even after the reactants 302and 304 have reacted with one another to generate the products 306 inthe central region. Similarly, because there are only reactants 304 inthe upper-left and lower-right regions, and none of the reactants 302 inthese regions, the reactants 304 in these regions remain even after thereactants 302 and 304 have reacted with one another to generate theproducts 306 in the central region.

Referring next to FIG. 4, there are still two sets of reactants 302 and304 within the biochemical system 300, denoted as solid circles andhollow circles, respectively. However, it is presumed that the reactants302 are homogenously, or uniformly, distributed over a single spatialregion 402, and similarly it is presumed that the reactants 304 arehomogenously, or uniformly, distributed over the same spatial region402. The reaction of the reactants 302 and 304 again results in the setof the products 306, which are denoted as small X's.

However, in FIG. 4 all of the reactants 302 and 304 have been consumedto generate the products 306. This is because of the spatial homogeneityassumption. That is, since there are sufficient numbers of both thereactants 302 and 304 in FIG. 4, all possible reactions between thesetwo reactants 302 and 304 resulting in the products 306 occur, becauseit is presumed that the reactants 302 and 304 are well mixed and thusspatially homogeneous within the spatial region 402. None of thereactants 302 and 304 thus remain upon generation of the products 306 inFIG. 4, in contradistinction to FIG. 3. Comparison of FIGS. 3 and 4therefore shows how assuming spatial homogeneity can significantlyaffect modeling of a biochemical system.

The modeling of the biochemical system 300 can be accomplished as inFIG. 3 by performing an embodiment of the invention, such as thatdescribed in subsequent sections of the detailed description. Themodeling of the system 300 as accomplished in FIG. 3 thus depicts theadvantages of the invention as compared to the prior art. Embodiments ofthe invention take into account the heterogeneously spatial nature ofbiochemical systems during modeling, as depicted in FIG. 3, whereasprior art approaches like that of [Goss and Peccoud] do not, as depictedin FIG. 4, and instead assume spatial homogeneity even where the systemin question is spatially heterogeneous.

Spatially Heterogeneous Stochastic Petri-Net Modeling

FIG. 5 shows a method 500 for spatially heterogeneous stochasticPetri-net modeling of a biochemical system, according to an embodimentof the invention. The method 500 leverages the spatially homogenousstochastic Petri-net modeling that has been described in relation toFIGS. 1 and 2. As an overview, the biochemical reactions of abiochemical system are assigned to different regions of a space to takeinto account the spatial heterogeneity of the biochemical system withinthe space. Thereafter, the biochemical reactions assigned to each regionare modeled as separate spatially homogenous stochastic Petri-nets, withinteractions between the regions modeled using advection, convection,and/or diffusion-based molecular movement.

The method 500 begins by defining the biochemical reactions of thebiochemical system (502). Definition of the biochemical reactions of thebiochemical system can in one embodiment be accomplished by performingthe method 600 of FIG. 6 for each biochemical reaction. The method 600is specifically for defining a stochastic Petri-net model for abiochemical reaction, such as a spatially homogenous stochasticPetri-net model. Referring now to FIG. 6, the number of places of astochastic Petri-net for a biochemical reaction is defined (602). Thetokens that initially populate the places are also defined (604). Eachtoken corresponds to a reactant, such as a molecule, within thebiochemical reaction, and is initially located in one of the places ofthe stochastic Petri-net that have been defined.

The method 600 next defines the transitions of the stochastic Petri-netfor the biochemical reaction (606). Each transition corresponds to areaction, as has been described in relation to FIG. 1. The method 600further defines a label for each transition (608). Each labelcorresponds to the number of time steps that are to occur betweensuccessive firings of its corresponding transition. A label of zeroindicates that a number of such firings can occur instantaneously,insofar as there are sufficient tokens to be input to and consumed bythe corresponding transition.

One or more input/output functions of the stochastic Petri-net are thendefined (608). Each input/output function links an input place ortransition to an output place or transition, as a directed arc, as hasbeen described in relation to FIG. 1. Furthermore, a weight is definedfor each input/output function (610). The weight of an input/outputfunction corresponds to the number of tokens required for theinput/output function to occur.

Each transition thus has one or more input/output functions leading froma place to the transition. The transition is enabled when each placeleading to the transition via an input/output function has a number oftokens located therein greater than or equal to the weight specified bythe input/output function in question. Furthermore, at least one of theplaces can be an output place. An output place has one or moreinput/output functions leading to the output place, and leading fromanother place or a transition. The occurrence of each input/outputfunction of the output place results in transfer of tokens from theother place or the transition leading from the input/output function tothe output place. In FIG. 1, the place 102 is an output place.

Referring back to FIG. 5, a spatial decomposition of the biochemicalsystem is next defined, such that the system is spatially heterogeneous(504). Definition of the spatial decomposition of the system ensuresthat the spatial heterogeneity of the biochemical system being modeledis taken into account. That is, the biochemical system is decomposedover a space, such that the system is spatially heterogeneous. Thedegree of precision to which the biochemical system is spatiallydecomposed corresponds to the desired accuracy of the modeling of thesystem. A complex spatial decomposition, for instance, likely results ingreater accuracy of the resulting modeling of the biochemical systemthan does a relatively simple spatial decomposition.

In one embodiment, spatial decomposition of the biochemical system isaccomplished as follows. First, the space within which the biochemicalsystem is to be modeled is divided into a number of regions, orcompartments, such as a number of orthants (506). Each biochemicalreaction defined in 502 is then assigned to one of these regions (508).A given region may encompass zero, one, or more than one of themolecular species involved in the biochemical reactions of the system,based on the molecules spatial distribution and the position of theregion in that space. Finally, the regions of the system are initiallypopulated with reactants (509), according to a desired initial moleculardistribution.

FIG. 7 shows an example spatial decomposition of a biochemical system,according to an embodiment of the invention, which may be performed asthe spatial decomposition in 504 of the method 500 of FIG. 5. Referringnow to FIG. 7, there is a cubic three-dimensional space 700 within whichthe biochemical system is to be modeled. This space 700 is divided intoa number of three-dimensional regions, or orthants, such as the regionor orthant 702, in 506 of the method 500 of FIG. 5. Each orthant is aunique three-dimensional sub-space of the three-dimensional space 700.In 508 of the method 500, then, each biochemical reaction is assigned toone of these regions.

As can be appreciated by those of ordinary skill within the art, thespatial decompositions of the biochemical system depicted in FIG. isjust one example of the kinds of spatial decompositions that can beperformed in 504 of the method 500 of FIG. 5. In other embodiments ofthe invention, other types of spatial decompositions may be employed.That is, embodiments of the invention are not limited to decompositionof a cubic space into orthants, as in FIG. 7.

Referring back to FIG. 5, the method 500 next defines relationships forinter-region movement of reactants among the regions into which thesystem has been spatially decomposed in 504 (510). Relationships amongthe regions can include flux, advection, convection, and/ordiffusion-based molecular movement for inter-region movement ofreactants of the biochemical reactions of the system (512), such asmolecules. Other types of relationships may also be defined for suchinter-region movement.

Referring back to FIG. 5, the method 500 next actually models thebiochemical system in question (514). The biochemical reactions of eachregion as have been defined are modeled as spatially homogenousstochastic Petri-nets (516), as has been described, for instance, inrelation to FIGS. 1 and 2. Inter-region movement of the reactants ofthese biochemical reactions are based on the input physical effects thatare present in the system and the physical laws that determine theirmotion, which are denoted herein as relationships for inter-regionmovement (518). In other words, the biochemical reactions of each regionare run as spatially homogenous stochastic Petri-nets, with thebiochemical reactions of the regions interacting with one another viathe inter-region movement of their reactants.

For example, the initial system is modeled as a number of discretehomogenous Petri-nets, which are populated by reactants in accordancewith an initial molecular distribution. In each time step, each of thePetri-nets within the regions is separately executed in a homogenousmanner, such that reactants are consumed, and products created, inaccordance with the biochemical reactions of the Petri-nets. However,between time steps, inter-region movement of reactants (i.e., molecules)occurs in accordance with relationships that have been defined, such asflux, advection, convection, and/or diffusion-based molecular movement.That is, based on a given inter-region movement model that has beenselected, reactants move among the regions. Therefore, the modeling of514, 516, and 518 is a process in which the Petri-nets are individuallyrun, then inter-region molecular movement occurs, the Petri-nets areagain individually run, inter-region molecular movement again occurs,and so on.

Heterogeneously spatial stochastic Petri-net modeling is thusaccomplished by leveraging spatially homogenous stochastic Petri-nets.Each biochemical reaction is modeled as a spatially homogenousstochastic Petri-net within its given region, and therefore can use thespatially homogenous stochastic Petri-net methodology that has beendescribed in relation to FIGS. 1 and 2. Inter-region movement is modeledas flux, advection, convection, and/or diffusion-based molecularmovement.

Once the biochemical system has been modeled—that is, once the modelingsimulation has run as long as desired—one or more actions may beperformed on the basis of that modeling (520). For instance, the resultsof the modeling of the biochemical system may be analyzed (522), such asby a scientist, to learn new insights into the functioning of thesystem. As another example, the biochemical system may itself beadjusted, modified, or manipulated based on the results of the modeling(524). For instance, drug development may entail adjusting the system tocorrespond to the effects of the revised formulation of a new drug, todetermine if the revised formulation has better results or efficacy thanan earlier formulation of the drug did.

Computerized System and Conclusion

FIG. 8 shows a rudimentary computerized system 1800 for heterogeneouslyspatial stochastic Petri-net modeling of a biochemical system, accordingto an embodiment of the invention. As depicted in FIG. 8, the system1800 includes one or more processors 1002, one or more computer-readablemedia 1004, and a computer program 1006. As can be appreciated by thoseof ordinary skill within the art, the system 1800 can, and typicallywill, include other components, in addition to and/or in lieu of thosedepicted in FIG. 8.

The computer-readable media 1004 may each be a volatile or non-volatilemedium, as well as a magnetic, optical, and/or semiconductor medium. Themedia 1004 store data representing the biochemical reactions 1008 of abiochemical system that have been described, the spatial regions 1010into which the biochemical system has heterogeneously spatiallydecomposed as has been described, and one or inter-region movementrelationships 1012 corresponding to potential inter-region movement ofreactants of the biochemical reactions 1008 as has also been described.Furthermore, the media 1004 store data representing the following foreach of the biochemical reactions 1008: places 1014 of a stochasticPetri-net for the biochemical reaction; tokens 1016 for the biochemicalreaction; transitions 1018 of the stochastic Petri-net; labels 1020 forthe transitions 1018; input/output functions 1022 of the stochasticPetri-net; and, weights 1024 for the input/output functions 1022. Eachof these data that the media 1004 store for each of the biochemicalreactions 1008 is as has been described in previous sections of thedetailed description.

The computer program 1006 is executed by the processors 1002, such asfrom the computer-readable media 1004 on which the computer program 1006may be stored. The computer program 1006 may be made up of one or morecomputer program parts, sections, objects, routines, and subroutines,among other types of computer program components, as can be appreciatedby those of ordinary skill within the art. The computer program 1006further may itself be made up of more than one computer program. Thecomputer program 1006 models the biochemical reactions 1008 of thebiochemical system using a heterogeneously spatial stochastic Petri-netapproach, as has been described in previous sections of the detaileddescription. That is, the computer program 1006 models the biochemicalreactions of each region as a spatially homogenous stochastic Petri-net,and further models inter-region movement of reactants based onrelationships such as flux, advection, convection, and/or diffusion, ashas been described.

It is noted that, although specific embodiments have been illustratedand described herein, it will be appreciated by those of ordinary skillin the art that any arrangement calculated to achieve the same purposemay be substituted for the specific embodiments shown. For example, thetypes of biochemical systems that can be modeled using the spatiallyheterogeneous stochastic Petri-net approach described herein are notlimited by embodiments of the invention. Such biochemical systems can beas diverse as the Hox gene system, for instance, which has a role inhindbrain development, as well as other biomolecular systems, and othertypes of biochemical systems, as can be appreciated by those of ordinaryskill within the art. The terminology biochemical system is inclusive ofany such type of biomolecular system. This application is thus intendedto cover any adaptations or variations of embodiments of the presentinvention. Therefore, it is manifestly intended that this invention belimited only by the claims and equivalents thereof.

1. A method, comprising: defining a biochemical reaction of a system,said biochemical reaction occurring between at least two reactants toform a product, said system comprising a space in which each reactant ofthe at least two reactants is non-uniformly spatially distributed;defining a spatial decomposition of the space by subdividing the spaceinto a plurality of regions, wherein each region of the plurality ofregions is a unique three-dimensional sub-space of a three-dimensionalspace; defining relationships for inter-region movement of the at leasttwo reactants between different regions of the plurality of regions,wherein defining relationships for inter-region movement of the at leasttwo reactants is selected from the group consisting of definingflux-based molecular movement, defining advection-based molecularmovement, defining convection-based molecular movement, definingdiffusion-based movement, and combinations thereof; modeling thebiochemical reaction in each region as a spatially homogenous stochasticPetri-net; modeling the inter-region movement of the at least tworeactants between said different regions based on the definedrelationships; determining, from said modeling the biochemical reactionin each region and said modeling the inter-region movement of the atleast two reactants between said different regions, an identification ofthe product of the biochemical reaction in each region; and storing theidentification of the product of the biochemical reaction in each regionin a computer readable medium, wherein said modeling the biochemicalreaction comprises modeling the biochemical reaction as occurring in asequence of time steps, wherein said modeling the biochemical reactionin each region as a spatially homogenous stochastic Petri-net isperformed in each time step, and wherein said modeling the inter-regionmovement of the at least two reactants between said different regionsbased on the defined relationships is performed between a first timestep and a second time step in each pair of successive time steps in thesequence of time steps, and wherein defining relationships forinter-region movement of the at least two reactants is selected from thegroup consisting of defining flux-based molecular movement, definingadvection-based molecular movement, defining convection-based molecularmovement, defining diffusion-based movement, and combinations thereof,and wherein said modeling the biochemical reaction in each region, saidmodeling the inter-region movement of the at least two reactants betweensaid different regions, said determining an identification of theproduct of the biochemical reaction in each region, and said storing theidentification of the product of the biochemical reaction in each regionare performed by a computer processor.
 2. The method of claim 1, whereinat least one reactant of the at least two reactants comprises amolecule.
 3. The method of claim 1, wherein defining the biochemicalreaction comprises: defining a plurality of places of a stochasticPetri-net for the biochemical reaction; defining a plurality of tokensfor the biochemical reaction, each token corresponding to a reactant ofthe at least two reactants and located in one place of the plurality ofplaces of the stochastic Petri-net for the biochemical reaction; and,defining one or more transitions of the stochastic Petri-net, eachtransition pertaining to the biochemical reaction.
 4. The method ofclaim 3, wherein each transition has one or more of input/outputfunctions, each input/output function having a weight and leading from aplace of the plurality of places and leading to a transition of the oneor more transitions, each transition being enabled when each placeleading to the transition via the input/output function has a number oftokens located therein greater than or equal to the weight of theinput/output function.
 5. The method of claim 3, wherein at least oneplace of the plurality of places is an output place having one or moreof the input/output functions each leading to the output place andleading from another place or a transition, such that occurrence of eachinput/output function of the output place results in transfer of tokensfrom the other place or the transition leading from the input/outputfunction to the output place.
 6. The method of claim 3, wherein themethod further comprises defining a global marking at a given time foreach biochemical reaction as a vector representing a number of tokens ineach place at the given time.
 7. The method of claim 1, wherein definingthe spatial decomposition of the space comprises defining a quad-treedecomposition of the space in accordance with a specified accuracy ofsaid modeling the biochemical reaction in each region.
 8. The method ofclaim 1, wherein defining the spatial decomposition of the spacecomprises defining the spatial decomposition of the space in accordancewith a specified accuracy of said modeling the biochemical reaction ineach region.
 9. The method of claim 1, wherein modeling the biochemicalreactions in each region comprises modeling the biochemical reaction ineach region as a spatially homogenous stochastic Petri-net, such thatthe biochemical reactions in the regions of the plurality of regionsinteract with one another via the inter-region movement of the at leasttwo reactants.
 10. The method of claim 1, said method further comprisingperforming one or more actions based on a result of said modeling thebiochemical reaction in each region.
 11. The method of claim 10, whereinsaid performing one or more actions comprises: analyzing results of saidmodeling the biochemical reaction in each region; and/or adjusting thesystem based on said results.